Puzzle:
Devise a fast algorithm for computing the number of 1-bits in an unsigned integer. If the machine supports n-bit integers, can we compute the number of 1-bits in O(log n) machine instructions? Can we compute the number of 1-bits in O(b) machine instructions where b is the number of 1-bits in the integer?

Source:
Commonly asked in job interviews for Software Engineers. I first heard it in 1996 when I was a graduate student.

Solution:
This article presents six solutions to this problem. Source code in C is available.

Iterated Count runs in time proportional to the total number of bits. It simply loops through all the bits, terminating slightly earlier because of the while condition. Useful if 1's are sparse and among the least significant bits.

Dense Ones runs in time proportional to the number of 0 bits. It is the same as Sparse Ones, except that it first toggles all bits (n ~= -1), and continually subtracts the number of 1 bits from sizeof(int).

Precompute_8bit assumes an array bits_in_char such that bits_in_char[i] contains the number of 1 bits in the binary representation for i. It repeatedly updates count by masking out the last eight bits in n, and indexing into bits_in_char.

Parallel Count carries out bit counting in a parallel fashion. Consider n after the first line has finished executing. Imagine splitting n into pairs of bits. Each pair contains the number of ones in those two bit positions in the original n. After the second line has finished executing, each nibble contains the number of ones in those four bits positions in the original n. Continuing this for five iterations, the 64 bits contain the number of ones among these sixty-four bit positions in the original n. That is what we wanted to compute.

Nifty Parallel Count works the same way as Parallel Count for the first three iterations. At the end of the third line (just before the return), each byte of n contains the number of ones in those eight bit positions in the original n. A little thought then explains why the remainder modulo 255 works.

MIT HAKMEM Count is funky. Consider a 3 bit number as being 4a+2b+c. If we shift it right 1 bit, we have 2a+b. Subtracting this from the original gives 2a+b+c. If we right-shift the original 3-bit number by two bits, we get a, and so with another subtraction we have a+b+c, which is the number of bits in the original number. How is this insight employed? The first assignment statement in the routine computes tmp. Consider the octal representation of tmp. Each digit in the octal representation is simply the number of 1's in the corresponding three bit positions in n. The last return statement sums these octal digits to produce the final answer. The key idea is to add adjacent pairs of octal digits together and then compute the remainder modulus 63. This is accomplished by right-shifting tmp by three bits, adding it to tmp itself and ANDing with a suitable mask. This yields a number in which groups of six adjacent bits (starting from the LSB) contain the number of 1's among those six positions in n. This number modulo 63 yields the final answer. For 64-bit numbers, we would have to add triples of octal digits and use modulus 1023. This is HACKMEM 169, as used in X11 sources. Source: MIT AI Lab memo, late 1970's.

8. Builtin Instructions

GNU compiler allows for

int __builtin_popcount (unsigned int x);

which translates into a single CPU instruction if the underlying machine architecture supports it. For example, Intel machines have POPCNT (SSE4 Instruction set announced in 2006). Many GCC builtin functions exist.

Which of the several bit counting routines is the fastest? Results of speed trials on an i686 are summarized in the table on left. "No Optimization" was compiled with plain gcc. "Some Optimizations" was gcc -O3. "Heavy Optimizations" corresponds to gcc -O3 -mcpu=i686 -march=i686 -fforce-addr -funroll-loops -frerun-cse-after-loop -frerun-loop-opt -malign-functions=4.

Thanks to Seth Robertson who suggested performing speed trials by extending bitcount.c. Seth also pointed me to MIT_Hackmem routine. Thanks to Denny Gursky who suggested the idea of Precompute_11bit. That would require three sums (11-bit, 11-bit and 10-bit precomputed counts). I then tried Precompute_16bit which turned out to be even faster.

If you have niftier solutions up your sleeves, please send me an e-mail or write comments below!